34 ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID [354
so that
^* ^t
dx J o
\/t
o (t — ry • ° o o
- r2) rdr = TT {<£ («) - 0 (0)},
o
where r2 = x2 + 2/2.
Now, if t' be any time between. 0 and t, we have, as in (19),
Multiplying this by (t — t')~^ dt' and integrating between 0 and t, we get
I ' V'MM + ?.^ f ' dt> !' V'^dr •'« (^-0* ^ ^ (t-rflo (t'-r)^
= ..... (22)
'
In (22) the first integral is the same as the integral in (19). By Abel's theorem the double integral in (22) is equal to TrV(t), since F(0)=0. Thus
(23)
If we now eliminate the integral between (19) and (23), we obtain simply
dV 4>p2v Tr kpv- . ._,..
____ ' I/ — r, _ i n , I4- I O/L\
~77 „ ' — a -----1" y \ " '....................V /
as the differential equation governing the motion of the lamina.
This is a linear equation of the first order. Since V vanishes with t, the integral may be written
in which t' = t. 4p2j//<r2. When i, or t', is great,
sotbat = _1+ ! +... ............. (27)
g'o-2 VTT v ^ '
Ultimately, when t is very great,n ingenious application of Abel's theorem Boggio has succeeded in integrating equations which include (19)*. The theorem is as follows: — If ty(t) be defined by